ArgyrisPack

Overview

ArgyrisPack is a library for doing things with the Argyris-5 finite element, a C1 finite element with 21 degrees of freedom. The Argyris-5 element spans the space of 5th degree polynomials on each element and has O(h^4) convergence in the H2 norm. The main goals of ArgyrisPack are numerical correctness, performance, and portability. ArgyrisPack is mostly written in C and can be called from C, Matlab, Julia, or Python.

ArgyrisPack is available under the BSD-3 License. See the file License.txt for more details.

Design & Usage Philosophy

ArgyrisPack assumes that it is faster to work with precomputed values of reference basis functions. Therefore it is up to the user to set up appropriate tables of reference data (function values, derivative values, presumably second derivative values) and to use these when setting up the typical per-element matricies.

Since performance and portability are both goals, the C code does not offer much abstraction; one passes in pointers to arrays and the functions fill the arrays appropriately. The non-C interfaces take care of wrapping output values and checking bounds instead.

Compilation Details

ArgyrisPack is written in C99 and may be compiled to use either row-major or column-major storage. Macros (contained in order_logic.h) ensure that the C code does 'the right things' (correct BLAS calls, correct order traversal, etc) in both cases. This is done by passing either -DUSE_ROW_MAJOR or -DUSE_COL_MAJOR to the compiler. In particular, with the makefile, compile the library with

make STORAGE_ORDER=USE_ROW_MAJOR

to pass one symbol or another. To compile the mex-files, run

make

at the MATLAB prompt. MATLAB (to the best of my knowledge) assumes MEX files on *nix platforms will use the gcc, but you may need to tweak your local MEX settings to compile the binaries correctly. Julia just needs libargyris_pack.so in the current path to work. Python needs for the .so to be in the working path as well.

ArgyrisPack currently targets GNU/Linux and OS X. GCC 4.2 and later should work; please let us know if this is not the case. In principle, there is nothing preventing a port to other operating systems, but all the provided makefiles assume a *nix environment.

What can ArgyrisPack do?

ArgyrisPack consists of four levels of C and Python functions/classes:

• Level 1: evaluation of the Argyris basis functions (and derivatives) at arbitrary points on the reference triangle. These are the functions beginning with ref_.
• Level 2: evaluation of the required coordinate transformations (Dominguez and Sayas' C, B, b, and Theta (Th) matricies) and evaluation of Argyris basis functions on arbitrary triangles. These are the functions beginning with physical_.
• Level 3: evaluation of a few common bilinear forms (the classic stiffness and mass matricies as well as the 'biharmonic' matrix resulting from discretization of the biharmonic operator). These functions begin with matrix_.
• Level 4: mesh generation. Argyris elements have 21 nodes (5 on each corner and one at the midpoint of each triangle edge). ArgyrisPack contains mesh parsing and creation classes for a variety of textual representations of meshes.

There are a few additional files; we wrote a 'multiply by a diagonal matrix' routine, wrappers to make dgemm work with row or column order, as well as a symbolic (symbolic.py and symbolic.m) version of the Argyris element to verify the numerical computations.

Running the Tests

The tests are currently implemented in Python. They require SAGE to be installed and used as a library. This is because the tests work by interpolating the Argyris polynomials on a physical element and then comparing the symbolic results with the numerical results. To run the results from SAGE, go to the root directory of ArgyrisPack and execute

import ap.test; ap.test.run()

The meshing software has tests, but these are not linked to the numerical tests at the moment.

Authors

• Erich Foster, Virginia Tech
• Traian Iliescu, Virginia Tech
• David Wells, Virginia Tech

Relevant Papers

ArgyrisPack is a new implementation of older ideas. We are greatly indebted to those who have done the necessary mathematical research for making this library possible.

• Dominguez and Sayas: A simple Matlab implementation of the Argyris element, 2006
• Argyris, Fried and Scharpf: The TUBA Family of Plate Elements for the Matrix Displacement Method, 1968.